Clustering and classification

This chapter will teach you how one can cluster and classify datasets. There is different methods to perform the clustering of the data, but in a very simple point of view you cluster what is similar. For that you can use different parameters, from simple concepts of means to more complex concepts like data populations distances.

About this week working data set “Boston”

The data set Boston is included in the library {MASS} and refers to the “Housing Values in Suburbs of Boston”. The data compiled in the dataset orginates from the following papers: Harrison and Rubinfeld (1978) & Belsley et al 1980

The data to be used in this exercise will be loaded from the library {MASS}.

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The Boston data set includes the Housing Values in Suburbs of Boston (from 1978 - 1980). The information included in this data set includes values regarding Boston city crime, nitrogen oxide concentrations, average rooms per dwelling, lower status of the population, pupil-teacher ratio in town, etc..

## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
## [1] 506  14

You can see that the data includes a total of 506 observations distributed into the following 14 variables: - crim = per capita crime rate by town. - zn = proportion of residential land zoned for lots over 25,000 sq.ft. - indus = proportion of non-retail business acres per town. - chas = Charles River dummy variable (= 1 if tract bounds river; 0 otherwise). - nox = nitrogen oxides concentration (parts per 10 million). - rm = average number of rooms per dwelling. - age = proportion of owner-occupied units built prior to 1940. - dis = weighted mean of distances to five Boston employment centres. - rad = index of accessibility to radial highways. - tax = full-value property-tax rate per $10,000. - ptratio = pupil-teacher ratio by town. - black = 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town. - lstat = lower status of the population (percent). - medv = median value of owner-occupied homes in $1000s.

Lets have a graphical overview of the data

  • From the graphical overview using the ggpairs between the variables you can start by observing which of the variables are normally distributed and which of them presented skewed data. In this data set most of the variables presented themselves skewed in either direction, being the rm (average number of rooms per dwelling) the only variable that presents a normal distribution.

For a better understanding of the data lets also have a look on the summaries of the variables included in this dataset.

##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08204   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00
  • From the variables summary you can mainly check the ranges of the data in the different variables. The skweded distrubition of most variables is now noticeable when you compare the Mean values and the minimum and maximum values of the data.

For a better understanding on how the variables relate to each other, lets visualize a correlation plot.

## corrplot 0.84 loaded
##          crim    zn indus  chas   nox    rm   age   dis   rad   tax
## crim     1.00 -0.20  0.41 -0.06  0.42 -0.22  0.35 -0.38  0.63  0.58
## zn      -0.20  1.00 -0.53 -0.04 -0.52  0.31 -0.57  0.66 -0.31 -0.31
## indus    0.41 -0.53  1.00  0.06  0.76 -0.39  0.64 -0.71  0.60  0.72
## chas    -0.06 -0.04  0.06  1.00  0.09  0.09  0.09 -0.10 -0.01 -0.04
## nox      0.42 -0.52  0.76  0.09  1.00 -0.30  0.73 -0.77  0.61  0.67
## rm      -0.22  0.31 -0.39  0.09 -0.30  1.00 -0.24  0.21 -0.21 -0.29
## age      0.35 -0.57  0.64  0.09  0.73 -0.24  1.00 -0.75  0.46  0.51
## dis     -0.38  0.66 -0.71 -0.10 -0.77  0.21 -0.75  1.00 -0.49 -0.53
## rad      0.63 -0.31  0.60 -0.01  0.61 -0.21  0.46 -0.49  1.00  0.91
## tax      0.58 -0.31  0.72 -0.04  0.67 -0.29  0.51 -0.53  0.91  1.00
## ptratio  0.29 -0.39  0.38 -0.12  0.19 -0.36  0.26 -0.23  0.46  0.46
## black   -0.39  0.18 -0.36  0.05 -0.38  0.13 -0.27  0.29 -0.44 -0.44
## lstat    0.46 -0.41  0.60 -0.05  0.59 -0.61  0.60 -0.50  0.49  0.54
## medv    -0.39  0.36 -0.48  0.18 -0.43  0.70 -0.38  0.25 -0.38 -0.47
##         ptratio black lstat  medv
## crim       0.29 -0.39  0.46 -0.39
## zn        -0.39  0.18 -0.41  0.36
## indus      0.38 -0.36  0.60 -0.48
## chas      -0.12  0.05 -0.05  0.18
## nox        0.19 -0.38  0.59 -0.43
## rm        -0.36  0.13 -0.61  0.70
## age        0.26 -0.27  0.60 -0.38
## dis       -0.23  0.29 -0.50  0.25
## rad        0.46 -0.44  0.49 -0.38
## tax        0.46 -0.44  0.54 -0.47
## ptratio    1.00 -0.18  0.37 -0.51
## black     -0.18  1.00 -0.37  0.33
## lstat      0.37 -0.37  1.00 -0.74
## medv      -0.51  0.33 -0.74  1.00

  • By drawing the correlation plot you can see some variables are highly positive or negatively correlated with each other.

  • High positive numbers (Blue) mean that variables value will shift together in the same direction. For example, when you look at the crime data, you can see its higly positvely correlated with the tax, rad and lstat variables. Meaning that the higher the index of acessibility to radial highways, the higher value of property-tax rate per $10.000, and the amount of low status of the population the bigger amount of crime you can witness on those areas.

  • On the other side, you can also observe some High negative numbers (Red) meaning that while the value of one variable increases the other one will decrease. For example, you can see how lstat (percentage of the lower status of the population) inversely correlates with medv (median value of owner-occupied homes in $1000s) and rm (average number of rooms per dwelling). Meaning that the higher amount of low status population the less amount of rooms per dwelling, and the less amount of owner occupied homes with the value of $1000.

Standardizing the dataset and explore the scaled data summary and how variables change.

##       crim                 zn               indus        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202  
##       chas              nox                rm               age         
##  Min.   :-0.2723   Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331  
##  1st Qu.:-0.2723   1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366  
##  Median :-0.2723   Median :-0.1441   Median :-0.1084   Median : 0.3171  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.:-0.2723   3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059  
##  Max.   : 3.6648   Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164  
##       dis               rad               tax             ptratio       
##  Min.   :-1.2658   Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047  
##  1st Qu.:-0.8049   1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876  
##  Median :-0.2790   Median :-0.5225   Median :-0.4642   Median : 0.2746  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6617   3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058  
##  Max.   : 3.9566   Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372  
##      black             lstat              medv        
##  Min.   :-3.9033   Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.: 0.2049   1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median : 0.3808   Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4332   3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 0.4406   Max.   : 3.5453   Max.   : 2.9865
## [1] "matrix"
  • When scaling the data, you just change the scale from which the data distributes in the x axis, not changing the values of the data itself. One can think this process is like grabbing the axis at the two ends and stretching or compressing it in order to make all your variables vary in the same comparable range, making it easier to compare. The scaling takes in account the mean (average) and standard deviation of the data.

Create a categorical variable of the crime rate in the Boston dataset (from the scaled crime rate). Quantiles from the old “crim”" variable will be used as the break points for the new “crime”" categorical variable.

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -0.419367 -0.410563 -0.390280  0.000000  0.007389  9.924110
##           0%          25%          50%          75%         100% 
## -0.419366929 -0.410563278 -0.390280295  0.007389247  9.924109610
## crime
##      low  med_low med_high     high 
##      127      126      126      127
  • The new crime variable is now easier to interpret the amount of data fiiting in the the new assigned labels: “low”, “med_low”, “med_high”, “high”. You can also verify how the distribution between this categories is quite even.

Since the new crime category looks more informative, let’s drop the old crim variable from the dataset.

## # A tibble: 506 x 14
## # Groups:   crime [4]
##             zn      indus       chas        nox         rm         age
##  *       <dbl>      <dbl>      <dbl>      <dbl>      <dbl>       <dbl>
##  1  0.28454827 -1.2866362 -0.2723291 -0.1440749  0.4132629 -0.11989477
##  2 -0.48724019 -0.5927944 -0.2723291 -0.7395304  0.1940824  0.36680343
##  3 -0.48724019 -0.5927944 -0.2723291 -0.7395304  1.2814456 -0.26554897
##  4 -0.48724019 -1.3055857 -0.2723291 -0.8344581  1.0152978 -0.80908783
##  5 -0.48724019 -1.3055857 -0.2723291 -0.8344581  1.2273620 -0.51067434
##  6 -0.48724019 -1.3055857 -0.2723291 -0.8344581  0.2068916 -0.35080997
##  7  0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.3880270 -0.07015919
##  8  0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.1603069  0.97784057
##  9  0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.9302853  1.11638970
## 10  0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.3994130  0.61548134
## # ... with 496 more rows, and 8 more variables: dis <dbl>, rad <dbl>,
## #   tax <dbl>, ptratio <dbl>, black <dbl>, lstat <dbl>, medv <dbl>,
## #   crime <fctr>
  • Now you can easily explore how the other variables vary according to the high or low crime rate category, and to have a better understanding on that I group the variables by crime categories.

Divide the dataset to train and test sets, so that 80% of the data belongs to the train set.

## [1] 506
##              zn       indus       chas         nox           rm
## 223 -0.48724019 -0.71961001  3.6647712 -0.41159834  0.845930986
## 419 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.466305712
## 433 -0.48724019  1.01499462 -0.2723291  0.25289548  0.199775388
## 240  0.79907391 -0.90473168 -0.2723291 -1.09335175  0.457383677
## 459 -0.48724019  1.01499462 -0.2723291  1.36613839  0.023292362
## 303  0.97058245 -0.73564417 -0.2723291 -1.05020280  0.299402903
## 61   0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.773727758
## 212 -0.48724019 -0.07970124  3.6647712 -0.56693456 -1.253363081
## 352  2.08538800 -1.37701059 -0.2723291 -1.24005818  0.418955921
## 376 -0.48724019  1.01499462 -0.2723291  1.00368721  1.463621579
## 280  0.37030254 -1.13795583 -0.2723291 -0.96476788  0.750573221
## 56   3.37170210 -1.44552019 -0.2723291 -1.30909650  1.372533565
## 189  1.44223095 -1.12192167 -0.2723291 -1.01568364  0.386221166
## 262  0.37030254 -1.04466617 -0.2723291  0.79657225  1.758234373
## 294 -0.48724019  0.40569652 -0.2723291 -1.01568364 -0.224353176
## 485 -0.48724019  1.01499462 -0.2723291  0.24426569 -0.588705230
## 241  0.79907391 -0.90473168 -0.2723291 -1.09335175  0.871549489
## 210 -0.48724019 -0.07970124  3.6647712 -0.56693456 -1.338758093
## 25  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.513272969
## 492 -0.48724019  2.42017014 -0.2723291  0.46864023 -0.429301206
## 172 -0.48724019  1.23072696 -0.2723291  0.43412107 -0.575895979
## 87  -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.383757200
## 244  0.79907391 -0.90473168 -0.2723291 -1.09335175  0.154231381
## 276  1.22784527 -0.68899934 -0.2723291 -0.92938574  0.810349730
## 277  1.22784527 -0.68899934  3.6647712 -0.92938574  1.398152069
## 389 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.999146193
## 410 -0.48724019  1.01499462 -0.2723291  0.36508275  0.807503230
## 182 -0.48724019 -1.26477147 -0.2723291 -0.57556435 -0.200157922
## 300  2.51415937 -1.29683979 -0.2723291 -1.33498587  1.076497520
## 206 -0.48724019 -0.07970124 -0.2723291 -0.56693456 -0.560240226
## 211 -0.48724019 -0.07970124  3.6647712 -0.56693456 -0.462035961
## 288  1.76380948 -0.84788329 -0.2723291 -1.29183692 -0.107646658
## 367 -0.48724019  1.01499462 -0.2723291  1.40928733 -1.881016425
## 95   0.71331963  0.56895343 -0.2723291 -0.78267931 -0.050716649
## 220 -0.48724019  0.40132357  3.6647712 -0.04051738  0.125766377
## 349  2.94293073 -1.33036576 -0.2723291 -1.03294322  0.498657933
## 353  2.08538800 -1.37701059 -0.2723291 -1.24005818 -0.570202978
## 266  0.37030254 -1.04466617 -0.2723291  0.79657225 -1.031336047
## 361 -0.48724019  1.01499462 -0.2723291  1.85803641  0.161347633
## 270  0.37030254 -0.60882854  3.6647712 -0.78267931 -0.518965970
## 396 -0.48724019  1.01499462 -0.2723291  1.19354259  0.265244898
## 381 -0.48724019  1.01499462 -0.2723291  1.00368721  0.972600255
## 292  2.94293073 -0.90181638 -0.2723291 -1.24005818  1.228785293
## 81   0.58468822 -0.91493524 -0.2723291 -1.11061133  0.629596953
## 328 -0.48724019 -0.54760720 -0.2723291 -0.53241540 -0.286976185
## 374 -0.48724019  1.01499462 -0.2723291  0.97779784 -1.962141687
## 460 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.289822685
## 216 -0.48724019 -0.07970124 -0.2723291 -0.56693456 -0.146074414
## 165 -0.48724019  1.23072696 -0.2723291  0.43412107 -0.612900484
## 382 -0.48724019  1.01499462 -0.2723291  1.00368721  0.370565414
## 264  0.37030254 -1.04466617 -0.2723291  0.79657225  1.483547082
## 426 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.553123975
## 66   2.94293073 -1.13212523 -0.2723291 -1.35224545  0.007636609
## 255  2.94293073 -1.09276866 -0.2723291 -1.40402419 -0.251394930
## 36  -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.500463717
## 153 -0.48724019  1.23072696  3.6647712  2.72964520 -1.811277165
## 128 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.842043769
## 53   0.41317968 -0.80123846 -0.2723291 -0.99842406  0.322174907
## 228 -0.48724019 -0.71961001 -0.2723291 -0.43748771  1.250134047
## 13   0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.563086727
## 470 -0.48724019  1.01499462 -0.2723291  0.21837632 -0.813578764
## 425 -0.48724019  1.01499462 -0.2723291  0.25289548 -1.024219796
## 193  1.44223095 -1.12192167 -0.2723291 -1.01568364  1.271482800
## 136 -0.48724019  1.56744433 -0.2723291  0.59808708  0.071682869
## 100 -0.48724019 -1.20209248 -0.2723291 -0.94664532  1.610216351
## 301  2.51415937 -1.29683979 -0.2723291 -1.33498587  0.834544984
## 214 -0.48724019 -0.07970124 -0.2723291 -0.56693456  0.128612878
## 363 -0.48724019  1.01499462 -0.2723291  1.85803641 -1.313139590
## 377 -0.48724019  1.01499462 -0.2723291  1.00368721  0.518583436
## 480 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.079181654
## 68   0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.578742479
## 31  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.813578764
## 160 -0.48724019  1.23072696 -0.2723291  2.72964520  0.320751657
## 28  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.338213193
## 117 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.154613915
## 178 -0.48724019 -1.03300497 -0.2723291 -0.38570897  0.043217865
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## 476 -0.48724019  1.01499462 -0.2723291  0.25289548 -0.174539418
## 147 -0.48724019  1.23072696 -0.2723291  2.72964520 -0.934555033
## 327 -0.48724019 -0.54760720 -0.2723291 -0.53241540  0.038948114
## 346 -0.48724019 -0.98344483 -0.2723291 -0.97253469 -0.385180450
## 424 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.258511181
## 271  0.37030254 -0.60882854 -0.2723291 -0.78267931 -0.610053984
## 474 -0.48724019  1.01499462 -0.2723291  0.51178918  0.989679257
## 466 -0.48724019  1.01499462 -0.2723291  0.86561057 -0.748109254
## 289  1.76380948 -0.84788329 -0.2723291 -1.29183692  0.043217865
## 47  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.709681499
## 111 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.127572161
## 445 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.612900484
## 184 -0.48724019 -1.26477147 -0.2723291 -0.57556435  0.396183918
## 176 -0.48724019 -1.03300497 -0.2723291 -0.38570897  0.371988664
## 434 -0.48724019  1.01499462 -0.2723291  1.36613839  0.215431141
## 35  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 141 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.157460416
## 456 -0.48724019  1.01499462 -0.2723291  1.36613839  0.342100410
## 215 -0.48724019 -0.07970124 -0.2723291 -0.56693456 -1.241977079
## 96  -0.48724019 -1.20209248 -0.2723291 -0.94664532  0.484425431
## 464 -0.48724019  1.01499462 -0.2723291  1.36613839  0.325021407
## 226 -0.48724019 -0.71961001 -0.2723291 -0.43748771  3.473250881
## 116 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.507579968
## 137 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.487654465
## 202  3.05012357 -1.32745046 -0.2723291 -1.20553902 -0.174539418
## 293  2.94293073 -0.90181638 -0.2723291 -1.24005818  0.491541682
## 457 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.439263958
## 12   0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.392296701
## 304  0.97058245 -0.73564417 -0.2723291 -1.05020280  0.992525758
## 357 -0.48724019  1.01499462  3.6647712  1.85803641 -0.103376907
## 446 -0.48724019  1.01499462 -0.2723291  1.59914271  0.248165896
## 41   2.72854505 -1.19334657 -0.2723291 -1.09335175  1.052302267
## 237 -0.48724019 -0.71961001  3.6647712 -0.41159834  0.492964932
## 335 -0.48724019 -0.86683276 -0.2723291 -0.34256002  0.036101614
## 383 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.065494052
## 273  0.37030254 -0.60882854 -0.2723291 -0.78267931  0.360602663
## 315 -0.48724019 -0.18027916 -0.2723291 -0.09229612  0.401876919
## 272  0.37030254 -0.60882854 -0.2723291 -0.78267931 -0.063525901
## 461 -0.48724019  1.01499462 -0.2723291  1.36613839  0.592592447
## 395 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.565933227
## 414 -0.48724019  1.01499462 -0.2723291  0.36508275 -1.607752384
## 104 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.210120673
## 356  2.94293073 -1.34494227 -0.2723291 -1.22279860 -0.496193967
## 254  0.45605682 -0.76917014 -0.2723291 -1.06746238  2.810016281
## 275  1.22784527 -0.68899934  3.6647712 -0.92938574  0.673717710
## 102 -0.48724019 -0.37560439 -0.2723291 -0.29941107  0.706452465
## 484 -0.48724019  1.01499462 -0.2723291 -0.19585359 -0.743839504
## 475 -0.48724019  1.01499462 -0.2723291  0.25289548 -1.220628326
## 133 -0.48724019  1.56744433 -0.2723291  0.59808708  0.124343127
## 350  1.22784527 -1.44114723 -0.2723291 -1.08472196  0.931325998
## 45  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.306901688
## 62   0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.453496460
## 496 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.874778524
## 32  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.302631937
## 398 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.765188257
## 46  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.857699521
## 372 -0.48724019  1.01499462 -0.2723291  0.65849561 -0.097683907
## 234 -0.48724019 -0.71961001 -0.2723291 -0.41159834  2.792937279
## 400 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.615746985
## 80  -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.584435480
## 493 -0.48724019  2.42017014 -0.2723291  0.46864023 -0.429301206
## 418 -0.48724019  1.01499462 -0.2723291  1.07272553 -1.395688102
## 98  -0.48724019 -1.20209248 -0.2723291 -0.94664532  2.539598741
## 337 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.591551731
## 443 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.093414156
## 245  0.45605682 -0.76917014 -0.2723291 -1.06746238 -0.984368790
## 88  -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.232892677
## 302  0.97058245 -0.73564417 -0.2723291 -1.05020280  0.434611674
## 70   0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.568779727
## 429 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.130418661
## 489 -0.48724019  2.42017014 -0.2723291  0.46864023 -1.182200570
## 121 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.590128481
## 14  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.477691714
## 209 -0.48724019 -0.07970124  3.6647712 -0.56693456 -0.314017939
## 161 -0.48724019  1.23072696  3.6647712  0.43412107 -0.049293399
## 431 -0.48724019  1.01499462 -0.2723291  0.25289548  0.090185122
## 9    0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.930285282
## 473 -0.48724019  1.01499462 -0.2723291  0.21837632  0.216854391
## 179 -0.48724019 -1.03300497 -0.2723291 -0.38570897  0.818889232
## 4   -0.48724019 -1.30558569 -0.2723291 -0.83445805  1.015297761
## 54   0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.407952453
## 318 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -0.715374500
## 386 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.434115858
## 16  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.641365488
## 404 -0.48724019  1.01499462 -0.2723291  1.19354259 -1.331641842
## 281  0.37030254 -1.13795583 -0.2723291 -0.96476788  2.185209437
## 79  -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.074911903
## 175 -0.48724019 -1.03300497 -0.2723291 -0.38570897 -0.605784233
## 250  0.45605682 -0.76917014 -0.2723291 -1.06746238  0.616787701
## 185 -0.48724019 -1.26477147 -0.2723291 -0.57556435 -0.968713038
## 15  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 408 -0.48724019  1.01499462 -0.2723291  0.90012973 -0.963020037
## 3   -0.48724019 -0.59279438 -0.2723291 -0.73953036  1.281445551
## 345  1.87100232 -1.07236154 -0.2723291 -0.61008351  0.838814735
## 93   0.71331963  0.56895343 -0.2723291 -0.78267931  0.223970642
## 326 -0.48724019 -0.54760720 -0.2723291 -0.53241540  0.208314890
## 55   2.72854505 -1.04029322 -0.2723291 -1.24868797 -0.564509977
## 233 -0.48724019 -0.71961001 -0.2723291 -0.41159834  2.921029798
## 23  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.203004422
## 183 -0.48724019 -1.26477147 -0.2723291 -0.57556435  1.238748045
## 307  0.92770532 -1.30558569 -0.2723291 -0.71364099  1.615909352
## 336 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.352445695
##              age          dis        rad         tax     ptratio
## 223  0.324172928 -0.248345051 -0.1779443 -0.60068166 -0.48755665
## 419  1.116389695 -0.946209387  1.6596029  1.52941294  0.80577843
## 433  0.221149222 -0.757294495  1.6596029  1.52941294  0.80577843
## 240 -0.936979326  1.137315760 -0.4076377 -0.64221553 -0.85708096
## 459  0.537325421 -0.480570747  1.6596029  1.52941294  0.80577843
## 303 -1.782484217  0.805741156 -0.2927910 -0.47014661 -1.08803366
## 61  -0.084369352  1.629073854 -0.1779443 -0.73715011  0.57482574
## 212  0.711399958 -0.061757165 -0.6373311 -0.77868399  0.06672981
## 352 -1.160789444  3.284049986 -0.6373311  0.01639310 -0.07184181
## 376  1.041786323 -1.177152855  1.6596029  1.52941294  0.80577843
## 280 -1.292233482  0.145156405 -0.5224844 -1.14062207 -1.64232012
## 56  -1.658145262  2.327745519 -0.5224844 -1.08128796 -0.25660396
## 189 -1.402362270  0.366459420 -0.5224844 -0.06074124 -1.50374851
## 262  0.739820290 -0.786073385 -0.5224844 -0.85581834 -2.51994036
## 294 -1.782484217  0.810965047 -0.6373311 -0.70748306 -1.13422420
## 485 -0.947636951 -0.033738114  1.6596029  1.52941294  0.80577843
## 241 -0.507121797  1.206746019 -0.4076377 -0.64221553 -0.85708096
## 210  1.116389695  0.037971662 -0.6373311 -0.77868399  0.06672981
## 25   0.906789743  0.287103768 -0.6373311 -0.60068166  1.17530274
## 492  1.073759197 -0.915103491 -0.6373311  1.79641644  0.75958789
## 172  1.020471073 -0.667870981 -0.5224844 -0.03107419 -1.73470120
## 87  -0.833955621  0.300210985 -0.7521778 -0.95668632  0.02053927
## 244 -2.159053622  1.538890501 -0.4076377 -0.64221553 -0.85708096
## 276 -0.915664077  0.224274608 -0.6373311 -0.91515245 -0.39517558
## 277 -0.695406500  0.471174688 -0.6373311 -0.91515245 -0.39517558
## 389  1.116389695 -1.047410401  1.6596029  1.52941294  0.80577843
## 410  1.116389695 -1.106297898  1.6596029  1.52941294  0.80577843
## 182 -0.226471014 -0.568522074 -0.7521778 -1.27709053 -0.30279450
## 300 -2.080897708  1.915153117 -0.5224844 -0.29807769 -1.68851066
## 206 -1.643935096  0.071404563 -0.6373311 -0.77868399  0.06672981
## 211  0.835738912  0.038968950 -0.6373311 -0.77868399  0.06672981
## 288 -1.324206356  1.672669599 -0.4076377 -0.68374941 -0.85708096
## 367  0.810871121 -0.970096815  1.6596029  1.52941294  0.80577843
## 95   0.309962761 -0.085502124 -0.6373311 -0.82021787 -0.11803234
## 220  0.846396537 -0.205034246 -0.5224844 -0.78461740 -0.94946204
## 349 -1.381047021  2.160296071 -0.6373311 -0.76088376 -0.67231881
## 353 -1.778931675  3.284049986 -0.6373311  0.01639310 -0.07184181
## 266 -0.205155765 -0.858875429 -0.5224844 -0.85581834 -2.51994036
## 361  0.690084708 -0.606371538  1.6596029  1.52941294  0.80577843
## 270 -0.251338805  0.058154876 -0.7521778 -1.09908819  0.06672981
## 396  1.073759197 -0.982729133  1.6596029  1.52941294  0.80577843
## 381  0.828633829 -1.129567958  1.6596029  1.52941294  0.80577843
## 292 -1.452097852  0.627653966 -0.6373311 -0.96855315  0.34387304
## 81  -1.246050442  0.762525331 -0.6373311 -0.75495035  0.25149196
## 328 -0.883691203  0.769743799 -0.5224844 -0.71934988  0.52863520
## 374  1.116389695 -1.244636028  1.6596029  1.52941294  0.80577843
## 460  0.562193212 -0.511724132  1.6596029  1.52941294  0.80577843
## 216 -0.929874243  0.071404563 -0.6373311 -0.77868399  0.06672981
## 165  0.825081288 -0.652056838 -0.5224844 -0.03107419 -1.73470120
## 382  1.084416821 -1.080795813  1.6596029  1.52941294  0.80577843
## 264  0.920999910 -0.815042235 -0.5224844 -0.85581834 -2.51994036
## 426  0.952972784 -0.895395175  1.6596029  1.52941294  0.80577843
## 66  -1.803799466  1.337533251 -0.6373311 -0.42267932 -1.08803366
## 255 -1.299338565  2.576450217 -0.9818712 -0.55321437 -0.94946204
## 36  -0.013318520 -0.206458943 -0.5224844 -0.76681717  0.34387304
## 153  0.690084708 -1.037579988 -0.5224844 -0.03107419 -1.73470120
## 128  0.974288033 -0.953000445 -0.6373311  0.17066179  1.26768382
## 53  -1.686565595  1.434032764 -0.6373311 -0.98041997 -0.76469989
## 228  0.402328842 -0.275129364 -0.1779443 -0.60068166 -0.48755665
## 13  -1.050660656  0.786365270 -0.5224844 -0.57694801 -1.50374851
## 470 -0.421860800 -0.461289840  1.6596029  1.52941294  0.80577843
## 425  0.071942477 -0.822308192  1.6596029  1.52941294  0.80577843
## 193 -1.501833434  1.274989030 -0.5224844 -0.06074124 -1.50374851
## 136  1.052443947 -0.799892951 -0.6373311  0.17066179  1.26768382
## 100 -0.215813389 -0.142395045 -0.8670245 -0.78461740 -0.21041342
## 301 -0.752247165  1.915153117 -0.5224844 -0.29807769 -1.68851066
## 214 -1.288680940  0.071404563 -0.6373311 -0.77868399  0.06672981
## 363  0.981393116 -0.803264735  1.6596029  1.52941294  0.80577843
## 377  0.878369411 -1.163570739  1.6596029  1.52941294  0.80577843
## 480  0.690084708 -0.875639370  1.6596029  1.52941294  0.80577843
## 68  -1.675907970  1.283632195 -0.6373311 -0.37521203  0.20530143
## 31   0.906789743  0.207985566 -0.6373311 -0.60068166  1.17530274
## 160  1.116389695 -0.963638187 -0.5224844 -0.03107419 -1.73470120
## 28   0.718505041  0.312653344 -0.6373311 -0.60068166  1.17530274
## 117  0.139440767 -0.505740403 -0.4076377  0.14099473 -0.30279450
## 178  0.171413641 -0.226784628 -0.5224844 -0.66594918 -0.85708096
## 78  -0.809087830  0.140312434 -0.5224844 -0.06074124  0.11292035
## 34   0.938762618 -0.003724486 -0.6373311 -0.60068166  1.17530274
## 360  0.452064424 -0.610693120  1.6596029  1.52941294  0.80577843
## 191 -1.672355429  1.274989030 -0.5224844 -0.06074124 -1.50374851
## 482  0.224701764 -0.220041060  1.6596029  1.52941294  0.80577843
## 441  0.846396537 -0.916005799  1.6596029  1.52941294  0.80577843
## 127  0.960077867 -0.967769809 -0.8670245 -1.30675758  0.29768250
## 320 -0.347257427  0.098236367 -0.6373311 -0.61848189 -0.02565127
## 27   0.771793164  0.421215295 -0.6373311 -0.60068166  1.17530274
## 251 -1.974321461  1.710424083 -0.2927910 -0.46421320  0.29768250
## 39  -1.363284313  0.024816954 -0.5224844 -0.76681717  0.34387304
## 205 -1.302891107  0.628271335 -0.6373311 -1.09315478 -1.73470120
## 256 -1.757616426  2.576450217 -0.9818712 -0.55321437 -0.94946204
## 135  1.059549031 -0.688149176 -0.6373311  0.17066179  1.26768382
## 64  -0.894348827  1.987860180 -0.1779443 -0.73715011  0.57482574
## 451  0.853501620 -0.698786917  1.6596029  1.52941294  0.80577843
## 442  1.016918532 -0.821548354  1.6596029  1.52941294  0.80577843
## 355 -1.658145262  3.224877549 -0.6373311 -0.44047956  1.63720813
## 505  0.736267749 -0.667776001 -0.9818712 -0.80241764  1.17530274
## 40  -1.661697804  0.762715291 -0.7521778 -0.92701927 -0.07184181
## 359  0.526667797 -0.509254657  1.6596029  1.52941294  0.80577843
## 124  1.009813449 -0.878868684 -0.8670245 -1.30675758  0.29768250
## 164  0.899684660 -0.775530624 -0.5224844 -0.03107419 -1.73470120
## 30   0.665216917  0.210834961 -0.6373311 -0.60068166  1.17530274
## 497  0.153650933 -0.473209809 -0.4076377 -0.10227512  0.34387304
## 132  1.034681240 -0.723766614 -0.6373311  0.17066179  1.26768382
## 323 -0.663433626  0.439783853 -0.5224844 -0.71934988  0.52863520
## 85  -0.730931915  0.467470475 -0.7521778 -0.95668632  0.02053927
## 119  0.160756016 -0.625699934 -0.4076377  0.14099473 -0.30279450
## 447  0.988498199 -0.818271549  1.6596029  1.52941294  0.80577843
## 109  1.013365990 -0.646880437 -0.5224844 -0.14380900  1.12911220
## 432  0.913894827 -0.810578183  1.6596029  1.52941294  0.80577843
## 438  1.116389695 -0.893210639  1.6596029  1.52941294  0.80577843
## 313  0.775345706 -0.456398379 -0.6373311 -0.61848189 -0.02565127
## 174  0.551535588 -0.545536954 -0.5224844 -0.66594918 -0.85708096
## 74  -2.215894287  0.708671765 -0.6373311 -0.61254848  0.34387304
## 77   0.210491598  0.122123795 -0.5224844 -0.06074124  0.11292035
## 369  1.116389695 -1.169459489  1.6596029  1.52941294  0.80577843
## 115  0.555088129 -0.730652652 -0.4076377  0.14099473 -0.30279450
## 60  -0.759352248  1.489738437 -0.1779443 -0.73715011  0.57482574
## 444  1.116389695 -0.862722112  1.6596029  1.52941294  0.80577843
## 168  0.377461051 -0.650204732 -0.5224844 -0.03107419 -1.73470120
## 177 -0.759352248 -0.114043564 -0.5224844 -0.66594918 -0.85708096
## 217 -0.446728591 -0.324328918 -0.5224844 -0.78461740 -0.94946204
## 384  1.116389695 -1.074194714  1.6596029  1.52941294  0.80577843
## 173  0.707847416 -0.569376892 -0.5224844 -0.66594918 -0.85708096
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## 414  1.116389695 -1.047457891  1.6596029  1.52941294  0.80577843
## 104  0.668769459 -0.513053850 -0.5224844 -0.14380900  1.12911220
## 356 -1.743406260  3.224877549 -0.6373311 -0.44047956  1.63720813
## 254 -2.137738373  2.427521836 -0.2927910 -0.46421320  0.29768250
## 275 -1.267365691  0.134186234 -0.6373311 -0.91515245 -0.39517558
## 102  0.096810268 -0.445903107 -0.5224844 -0.14380900  1.12911220
## 484 -1.004477616  0.144016647  1.6596029  1.52941294  0.80577843
## 475  0.952972784 -0.648352625  1.6596029  1.52941294  0.80577843
## 133  1.041786323 -0.696982300 -0.6373311  0.17066179  1.26768382
## 350 -1.210525026  2.373098390 -0.9818712 -0.43454615  0.57482574
## 45  -1.015135240  0.914588047 -0.7521778 -1.03975408 -0.25660396
## 62   0.881921953  1.435837380 -0.1779443 -0.73715011  0.57482574
## 496 -1.413019895 -0.473209809 -0.4076377 -0.10227512  0.34387304
## 32   1.116389695  0.180441414 -0.6373311 -0.60068166  1.17530274
## 398  1.077311738 -1.026562327  1.6596029  1.52941294  0.80577843
## 46  -1.235392817  0.619913109 -0.7521778 -1.03975408 -0.25660396
## 372  1.116389695 -1.247058014  1.6596029  1.52941294  0.80577843
## 234  0.064837394 -0.067978344 -0.1779443 -0.60068166 -0.48755665
## 400  0.327725469 -1.089723917  1.6596029  1.52941294  0.80577843
## 80  -1.135921653  0.336018383 -0.5224844 -0.06074124  0.11292035
## 493  0.530220338 -0.800272871 -0.6373311  1.79641644  0.75958789
## 418  0.729162665 -1.019866249  1.6596029  1.52941294  0.80577843
## 98   0.263779721 -0.142395045 -0.8670245 -0.78461740 -0.21041342
## 337 -0.791325122  0.681982432 -0.5224844 -1.09315478  0.80577843
## 443  1.116389695 -0.850184774  1.6596029  1.52941294  0.80577843
## 245  0.281542429  1.975512802 -0.2927910 -0.46421320  0.29768250
## 88  -0.418308258 -0.022530493 -0.7521778 -0.95668632  0.02053927
## 302 -1.000925074  0.805741156 -0.2927910 -0.47014661 -1.08803366
## 70  -1.263813149  1.283632195 -0.6373311 -0.37521203  0.20530143
## 429  0.338383094 -0.883047797  1.6596029  1.52941294  0.80577843
## 489  0.857054162 -0.937518732 -0.6373311  1.79641644  0.75958789
## 121  0.039969603 -0.730082773 -0.8670245 -1.30675758  0.29768250
## 14  -0.240681180  0.433325224 -0.6373311 -0.60068166  1.17530274
## 209 -0.336599802  0.210929941 -0.6373311 -0.77868399  0.06672981
## 161  0.853501620 -0.948203963 -0.5224844 -0.03107419 -1.73470120
## 431  0.622586419 -0.827437103  1.6596029  1.52941294  0.80577843
## 9    1.116389695  1.086121629 -0.5224844 -0.57694801 -1.50374851
## 473  0.228254306 -0.426717180  1.6596029  1.52941294  0.80577843
## 179  0.206939056 -0.417789076 -0.5224844 -0.66594918 -0.85708096
## 4   -0.809087830  1.076671135 -0.7521778 -1.10502160  0.11292035
## 54  -1.675907970  1.434032764 -0.6373311 -0.98041997 -0.76469989
## 318  0.111020434  0.112388362 -0.6373311 -0.61848189 -0.02565127
## 386  1.048891406 -1.125008926  1.6596029  1.52941294  0.80577843
## 16  -0.428965883  0.334118787 -0.6373311 -0.60068166  1.17530274
## 404  0.974288033 -0.993604324  1.6596029  1.52941294  0.80577843
## 281 -0.144762558  0.427246515 -0.5224844 -1.14062207 -1.64232012
## 79  -0.528437047  0.578929311 -0.5224844 -0.06074124  0.11292035
## 175  0.004444187 -0.519132560 -0.5224844 -0.66594918 -0.85708096
## 250 -1.814457091  1.914535748 -0.2927910 -0.46421320  0.29768250
## 185  0.754030456 -0.383311396 -0.7521778 -1.27709053 -0.30279450
## 15   0.565745754  0.316689987 -0.6373311 -0.60068166  1.17530274
## 408  1.116389695 -1.191922219  1.6596029  1.52941294  0.80577843
## 3   -0.265548971  0.556609050 -0.8670245 -0.98635338 -0.30279450
## 345 -1.437887686  1.268150482 -0.5224844 -0.22687676 -0.39517558
## 93  -0.531989588 -0.061329756 -0.6373311 -0.82021787 -0.11803234
## 326 -1.913928255  0.769743799 -0.5224844 -0.71934988  0.52863520
## 55  -0.745142082  1.673856847 -0.7521778  0.36053095  1.22149328
## 233  0.167861099  0.020590352 -0.1779443 -0.60068166 -0.48755665
## 23   0.821528746  0.086363887 -0.6373311 -0.60068166  1.17530274
## 183  0.839291454 -0.519749928 -0.7521778 -1.27709053 -0.30279450
## 307  0.118125517 -0.330455118 -0.2927910 -1.10502160 -0.02565127
## 336 -1.210525026  1.040151389 -0.5224844 -1.09315478  0.80577843
##            black        lstat         medv    crime
## 223  0.369308489 -0.381324714  0.540082359 med_high
## 419 -3.726650277  1.115651608 -1.493166348     high
## 433 -2.833938506 -0.087250788 -0.699438243     high
## 240  0.296905730 -0.739814835  0.083416874  med_low
## 459 -0.925178346  0.500897066 -0.829914096     high
## 303  0.295043632 -0.557769070  0.420479494  med_low
## 61   0.421009097  0.069588640 -0.416740562  med_low
## 212  0.422433054  1.586169891 -0.351502635 med_high
## 352  0.154509988 -1.003081017  0.170400776      low
## 376  0.440615895  0.110198849 -0.819041108     high
## 280  0.440615895 -1.092703547  1.366429428  med_low
## 56   0.429990982 -1.098304955  1.399048391      low
## 189  0.286609423 -1.133313756  0.790161078  med_low
## 262  0.347182381 -0.755218707  2.236268447 med_high
## 294  0.440615895 -0.570372239  0.148654801  med_low
## 485  0.153962312  0.096195329 -0.210153795 med_high
## 241  0.378728515 -0.178273670 -0.057931966  med_low
## 210  0.440615895  1.461538560 -0.275391721 med_high
## 25   0.412465352  0.510699530 -0.753803182 med_high
## 492  0.366241503  0.758561840 -0.971262937  med_low
## 172 -0.093587210 -0.087250788 -0.373248611 med_high
## 87   0.430648193  0.028978431 -0.003567028      low
## 244  0.197557316 -1.045091578  0.126908825  med_low
## 276  0.440615895 -1.354569377  1.029366808  med_low
## 277  0.356821477 -0.924661303  1.159842661  med_low
## 389  0.177950518  2.516003640 -1.340944520     high
## 410 -1.942212553  0.998022037  0.540082359     high
## 182  0.440615895 -0.448541612  1.486032293      low
## 300  0.163272803 -1.108107419  0.703177176      low
## 206  0.440615895 -0.249691623  0.007305960  med_low
## 211  0.400635552  0.646533677 -0.090550930  med_low
## 288  0.440615895 -0.772022931  0.072543887      low
## 367 -0.445195159  0.188618563 -0.068804954     high
## 95   0.440615895 -0.288901480 -0.210153795      low
## 220  0.406002776 -0.301504649  0.050797911  med_low
## 349  0.375332924 -0.933063415  0.213892727      low
## 353  0.390558315 -0.681000049 -0.427613550      low
## 266  0.391325061 -0.308506409  0.029051936 med_high
## 361  0.195914288 -0.681000049  0.268257666     high
## 270  0.379714331  0.139606242 -0.199280807  med_low
## 396  0.386724583  0.625528397 -1.025627875     high
## 381  0.440615895  0.638131565 -1.319198544     high
## 292  0.440615895 -1.273348959  1.605635158      low
## 81   0.440615895 -1.031088057  0.594447298      low
## 328  0.440615895  0.019175967 -0.036185991  med_low
## 374  0.440615895  3.097149734 -0.949516961     high
## 460  0.440615895  0.286643205 -0.275391721     high
## 216  0.404797889 -0.445740908  0.268257666  med_low
## 165  0.421009097 -0.141864517  0.018178948 med_high
## 382  0.440615895  1.180067802 -1.264833606     high
## 264  0.402497650 -0.196478246  0.920636930 med_high
## 426 -3.822712635  1.643584324 -1.547531287     high
## 66   0.440615895 -1.117909883  0.105162850      low
## 255  0.396692285 -0.851842997 -0.068804954      low
## 36   0.440615895 -0.416333515 -0.394994586      low
## 153 -0.146711775 -0.074647619 -0.786422145 med_high
## 128  0.388148541  0.635330861 -0.688565255 med_high
## 53   0.440615895 -1.032488409  0.268257666      low
## 228  0.168749562 -0.881250390  0.985874857 med_high
## 13   0.370513376  0.428078760 -0.090550930  med_low
## 470  0.440615895  0.295045317 -0.264518733     high
## 425 -3.866855315  0.631129805 -1.177849704     high
## 193  0.370403840 -1.369973249  1.507778268  med_low
## 136  0.416189548  0.603122764 -0.481978488 med_high
## 100  0.440615895 -0.905056374  1.159842661      low
## 301  0.374456642 -0.921860599  0.246511690      low
## 214  0.319141373 -0.458344076  0.605320286  med_low
## 363  0.264154709 -0.344915562 -0.188407819     high
## 377  0.069510683  1.482543841 -0.938643973     high
## 480  0.291867112  0.063987232 -0.123169893     high
## 68   0.433057967 -0.637589136 -0.057931966      low
## 31   0.038293155  1.392921311 -1.069119826 med_high
## 160  0.083640722 -0.737014131  0.083416874 med_high
## 28  -0.550896614  0.647934029 -0.840787084 med_high
## 117  0.401183228 -0.085850436 -0.144915868  med_low
## 178  0.426376321 -0.891052854  0.224765715      low
## 78   0.331737920 -0.333712745 -0.188407819  med_low
## 34   0.022958229  0.797771697 -1.025627875 med_high
## 360  0.373142220  0.002371742  0.007305960     high
## 191  0.230089266 -1.057694746  1.573016195  med_low
## 482  0.398663919 -0.688001809  0.126908825     high
## 441  0.380919218  1.324304061 -1.308325557     high
## 127  0.028654058  2.045485358 -0.742930194 med_high
## 320  0.433277038  0.010773855 -0.166661844 med_high
## 27   0.221326452  0.302047077 -0.645073304 med_high
## 251  0.433824713 -0.945666583  0.203019739  med_low
## 39   0.402607185 -0.353317674  0.235638703  med_low
## 205  0.371061052 -1.368572897  2.986504601      low
## 256  0.421775843 -0.476548653 -0.177534831      low
## 135 -1.028689097  0.652135085 -0.753803182 med_high
## 64   0.426157250 -0.441539852  0.268257666  med_low
## 451 -3.903330533  0.670339662 -0.993008912     high
## 442  0.320784401  0.961612885 -0.590708366     high
## 355  0.286171282 -0.644590896 -0.471105500      low
## 505  0.402826256 -0.864446166 -0.057931966  med_low
## 40   0.426704926 -1.166922204  0.898890955      low
## 359  0.424514223 -0.164270149  0.018178948     high
## 124  0.149361834  1.786420232 -0.568962390  med_low
## 164  0.348058662 -1.306957408  2.986504601 med_high
## 30   0.258020739 -0.094252548 -0.166661844 med_high
## 497  0.440615895  1.188469914 -0.308010684 med_high
## 132  0.440615895 -0.055042691 -0.318883672 med_high
## 323  0.440615895 -0.693603218 -0.231899770 med_high
## 85   0.440615895 -0.424735627  0.148654801      low
## 119 -0.197645637  0.380466791 -0.231899770  med_low
## 447 -0.423507192  0.719351983 -0.829914096     high
## 109  0.422433054 -0.053642339 -0.297137697  med_low
## 432 -3.015985986  0.985418869 -0.916897998     high
## 438 -3.804748865  1.932056843 -1.504039336     high
## 313  0.434043784 -0.130661701 -0.340629648 med_high
## 174  0.425280969 -0.505956045  0.116035838  med_low
## 74   0.224502972 -0.716008850  0.094289862  med_low
## 77   0.186056121 -0.095652900 -0.275391721  med_low
## 369  0.206429666 -1.315359520  2.986504601     high
## 115  0.351235183 -0.308506409 -0.438486537  med_low
## 60   0.440615895 -0.480749709 -0.318883672  med_low
## 444  0.329218610  0.867789298 -0.775549157     high
## 168 -1.413705278 -0.071846915  0.137781813 med_high
## 177  0.400416482 -0.356118378  0.072543887      low
## 217  0.395706469  0.120001313  0.083416874      low
## 384  0.440615895  1.667390309 -1.112611777     high
## 173  0.440615895  0.285242853  0.061670899  med_low
## 229  0.228227168 -1.222936286  2.627696006 med_high
## 24   0.414765591  1.012025558 -0.873406047 med_high
## 299  0.126688050 -1.075899322 -0.003567028      low
## 122  0.229979731  0.226428068 -0.242772758      low
## 186  0.333380947  0.069588640  0.768415102      low
## 42   0.314759966 -1.094103899  0.442225470  med_low
## 279  0.440615895 -0.765021171  0.714050163      low
## 331  0.130302712 -0.498954285 -0.297137697      low
## 290  0.164806295 -0.440139500  0.246511690      low
## 155 -0.390537100  0.345457990 -0.601581353 med_high
## 190  0.440615895 -1.017084537  1.344683452  med_low
## 29   0.342472368  0.020576319 -0.449359525 med_high
## 287 -0.165113687  0.038780895 -0.264518733      low
## 103 -3.131326538 -0.283300072 -0.427613550  med_low
## 286  0.416737224 -0.619384560 -0.057931966      low
## 483  0.422871195 -0.790227508  0.268257666     high
## 407  0.148376017  1.496547361 -1.156103728     high
## 338  0.417723041 -0.293102536 -0.438486537      low
## 51   0.425938180  0.111599201 -0.308010684  med_low
## 393  0.440615895  1.824229736 -1.395309459     high
## 118  0.414436985 -0.329511689 -0.362375623  med_low
## 342  0.416956295 -1.003081017  1.105477722      low
## 50   0.440615895  0.496696010 -0.340629648  med_low
## 248  0.213220848 -0.350516970 -0.221026782  med_low
## 314  0.402169045 -0.665596177 -0.101423917 med_high
## 391  0.413560704  0.624128045 -0.808168120     high
## 321  0.440615895 -0.763620819  0.137781813  med_low
## 69   0.440615895  0.061186528 -0.558089402  med_low
## 399  0.440615895  2.511802584 -1.906339882     high
## 37   0.228774844 -0.174072614 -0.275391721  med_low
## 500  0.428238419  0.342657286 -0.547216415  med_low
## 258  0.361750561 -1.054894042  2.986504601 med_high
## 324  0.377414092 -0.127860997 -0.438486537 med_high
## 242  0.415641872 -0.035437762 -0.264518733  med_low
## 148  0.383767133  2.363365269 -0.862533059 med_high
## 83   0.440615895 -0.830837717  0.246511690      low
## 253  0.322208358 -1.277550015  0.768415102  med_low
## 156 -2.942816482  0.331454470 -0.753803182 med_high
## 387  0.440615895  2.188321265 -1.308325557     high
## 274  0.373470826 -0.850442645  1.377302416  med_low
## 67   0.440615895 -0.337913801 -0.340629648      low
## 366 -0.021622592 -0.774823635  0.540082359     high
## 114  0.440615895  0.621327341 -0.416740562  med_low
## 162  0.194490331 -1.529613381  2.986504601 med_high
## 312  0.440615895 -0.934463767 -0.047058979 med_high
## 73   0.375004318 -0.998879961  0.029051936  med_low
## 406  0.309940417  1.446134688 -1.906339882     high
## 322  0.440615895 -0.809832436  0.061670899  med_low
## 439 -3.151590547  2.992123331 -1.536658299     high
## 167  0.138298780 -1.253744030  2.986504601 med_high
## 218  0.395487398 -0.414933163  0.670558212      low
## 308  0.440615895 -0.717409202  0.616193274      low
## 472  0.423418871  0.030378783 -0.318883672     high
## 365 -0.023265620 -1.031088057 -0.068804954 med_high
## 120  0.381466894  0.134004834 -0.351502635  med_low
## 488  0.345539353 -0.168471205 -0.210153795     high
## 112  0.426266786 -0.349116618  0.029051936  med_low
## 113  0.419256534  0.498096362 -0.405867574  med_low
## 354  0.304354123 -1.141715868  0.822780041      low
## 334  0.361860096 -0.976474328 -0.036185991      low
## 11   0.392639483  1.091845624 -0.819041108  med_low
## 348  0.391653667 -0.881250390  0.061670899      low
## 340  0.440615895 -0.407931403 -0.384121599      low
## 2    0.440615895 -0.491952525 -0.101423917      low
## 57   0.440615895 -0.963871160  0.235638703      low
## 268  0.305230404 -0.730012370  2.986504601 med_high
## 449  0.440615895  0.766963952 -0.916897998     high
## 230  0.259225626 -1.245341918  0.975001869 med_high
## 332  0.409069761 -0.031236706 -0.590708366      low
## 7    0.426376321 -0.031236706  0.039924923  med_low
## 170 -0.291736362 -0.186675782 -0.025313003 med_high
## 198 -0.025894464 -0.566171183  0.844526016      low
## 203  0.423966547 -1.336364800  2.149284545      low
## 201  0.302601560 -1.148717628  1.127223698      low
## 188  0.407426733 -0.836439125  1.029366808      low
## 149  0.003460966  2.193922673 -0.514597451 med_high
## 204  0.395487398 -1.238340158  2.823409785      low
## 309  0.440615895 -1.136114460  0.029051936 med_high
## 22   0.392749018  0.164812578 -0.318883672 med_high
## 82   0.426704926 -0.760820115  0.148654801      low
## 97   0.385081555 -0.183875078 -0.123169893  med_low
## 259  0.291538506 -0.681000049  1.464286318 med_high
## 468 -0.278044464  1.213676250 -0.373248611     high
## 138  0.409726972  0.271239333 -0.590708366 med_high
## 481  0.440615895 -0.267896200  0.050797911     high
## 329  0.282228015 -0.375723306 -0.351502635      low
## 106  0.409398367  0.534505515 -0.329756660  med_low
## 63   0.440615895 -0.829437365 -0.036185991  med_low
## 199  0.389134357 -0.844841237  1.312064489      low
## 479  0.252215374  0.752960432 -0.862533059     high
## 125  0.248710248  0.689944590 -0.405867574  med_low
## 130  0.440615895  0.796371345 -0.895152022 med_high
## 416 -3.608352275  2.296148371 -1.667134152     high
## 380  0.406002776  1.278092444 -1.340944520     high
## 108  0.339733988  0.201221731 -0.231899770  med_low
## 171 -0.705231691  0.248833700 -0.558089402 med_high
## 437 -3.605723431  0.755761136 -1.406182446     high
## 252  0.223407620 -1.269147903  0.246511690  med_low
## 491 -0.418906714  2.384370549 -1.569277262  med_low
## 91   0.388915287 -0.538164142  0.007305960      low
## 75   0.418927928 -0.822435605  0.170400776      low
## 412 -3.522914830  1.199672730 -0.579835378     high
## 351  0.440615895 -0.934463767  0.039924923      low
## 18   0.329437681  0.282442149 -0.547216415 med_high
## 89   0.440615895 -1.001680665  0.116035838      low
## 283  0.223407620 -1.350368321  2.551585092      low
## 269  0.368322672 -1.329363040  2.279760398 med_high
## 196  0.411370000 -1.355969729  2.986504601      low
## 364 -0.039805433  0.278241093 -0.623327329     high
## 263  0.330642568 -0.944266231  2.856028748 med_high
## 126  0.310488093  0.302047077 -0.123169893  med_low
## 163  0.360764744 -1.503006692  2.986504601 med_high
## 131  0.420242350 -0.007430722 -0.362375623 med_high
## 465  0.440615895  0.079391104 -0.123169893     high
## 422 -0.401928760  0.426678408 -0.906025010     high
## 495  0.440615895  0.131204129  0.213892727 med_high
## 207  0.418380252 -0.235688103  0.203019739  med_low
## 285  0.416737224 -0.672597937  1.051112783      low
## 325  0.440615895 -0.914858839  0.268257666 med_high
## 151  0.176636095  0.202622083 -0.112296905 med_high
## 169 -0.652654802 -0.217483527  0.137781813 med_high
## 316  0.435358206 -0.161469445 -0.688565255  med_low
## 401  0.440615895  1.976868108 -1.841101956     high
## 76   0.290881295 -0.519959566 -0.123169893  med_low
## 247  0.367008250 -0.489151821  0.192146752 med_high
## 423 -0.713337295  0.202622083 -0.188407819     high
## 368 -2.467324237  0.094794977  0.061670899     high
## 296  0.440615895 -0.893853558  0.659685225  med_low
## 139  0.387381794  1.213676250 -1.003881900  med_low
## 371  0.387491330 -1.357370081  2.986504601     high
## 370  0.204348498 -1.249542974  2.986504601     high
## 235  0.038621760 -0.644590896  0.703177176 med_high
## 238  0.365803363 -1.109507771  0.975001869 med_high
## 134  0.350687507  0.332854822 -0.449359525 med_high
## 298  0.440615895  0.446283337 -0.242772758  med_low
## 440  0.440615895  1.432131167 -1.058246839     high
## 306  0.401840439 -0.521359918  0.637939249      low
## 197  0.440615895 -1.200530653  1.170715648      low
## 152 -0.165113687  0.087793216 -0.318883672 med_high
## 390  0.440615895  1.147859705 -1.199595679     high
## 224  0.440615895 -0.707606738  0.822780041 med_high
## 385 -0.775991422  2.517403992 -1.493166348     high
## 17   0.330533033 -0.850442645  0.061670899 med_high
## 105  0.394501581 -0.045240227 -0.264518733  med_low
## 358  0.379714331  0.086392864 -0.090550930     high
## 154 -1.037561447  0.439281577 -0.340629648 med_high
## 58   0.396801820 -1.218735230  0.985874857      low
## 129  0.440615895  0.383267495 -0.492851476 med_high
## 388  0.440615895  2.707851869 -1.645388177     high
## 208  0.358793111  0.757161488 -0.003567028  med_low
## 471  0.440615895  0.509299178 -0.286264709     high
## 221  0.383657598 -0.412132459  0.453098458 med_high
## 49   0.440615895  2.542610329 -0.884279035  med_low
## 144  0.440615895  1.927855787 -0.753803182     high
## 157 -2.936025300  0.488293898 -1.025627875 med_high
## 101  0.417175365 -0.452742668  0.540082359  med_low
## 462  0.380700148  0.279641445 -0.525470439     high
## 227  0.336338397 -1.333564096  1.638254121 med_high
## 222  0.422433054  1.233281179 -0.090550930 med_high
## 219  0.440615895  0.737556559 -0.112296905  med_low
## 494  0.440615895 -0.090051492 -0.079677942  med_low
## 159 -0.030494942 -0.871447926  0.192146752 med_high
## 140  0.440615895  0.813175569 -0.514597451 med_high
## 59   0.372485009 -0.811232788  0.083416874  med_low
## 463  0.440615895  0.187218211 -0.329756660     high
## 71   0.296358054 -0.830837717  0.181273764  med_low
## 278  0.402826256 -1.189327837  1.148969673      low
## 339  0.432291221 -0.580174703 -0.210153795      low
## 375  0.440615895  3.545262384 -0.949516961     high
## 397  0.440615895  0.940607604 -1.090865802     high
## 94   0.434372389 -0.902255670  0.268257666      low
## 467 -3.665748713  0.629729453 -0.384121599     high
## 257  0.324946738 -1.336364800  2.334125337      low
## 378  0.440615895  1.202473434 -1.003881900     high
## 38   0.440615895 -0.543765550 -0.166661844      low
## 65   0.400526017 -0.644590896  1.138096685      low
## 295  0.440615895 -0.315508169 -0.090550930      low
## 26  -0.583319029  0.540106923 -0.938643973 med_high
## 415 -2.939968567  3.406627533 -1.688880128     high
## 146 -2.012862748  2.121104367 -0.949516961 med_high
## 476 -0.590548351  1.602974115 -1.003881900     high
## 147 -2.052733556  0.559711851 -0.753803182 med_high
## 327  0.440615895 -0.910657783  0.050797911 med_high
## 346  0.317279275 -0.297303592 -0.547216415      low
## 424 -3.879232792  1.489545601 -0.993008912     high
## 271  0.350249366  0.048583360 -0.155788856 med_high
## 474  0.197228711 -0.139063813  0.790161078     high
## 466 -0.243979021  0.206823139 -0.286264709 med_high
## 289  0.440615895 -0.707606738 -0.025313003      low
## 47   0.440615895  0.209623843 -0.275391721  med_low
## 111  0.403264396  0.048583360 -0.090550930  med_low
## 445 -1.272295352  1.559563202 -1.275706593     high
## 184  0.440615895 -0.976474328  1.083731747  med_low
## 176  0.375551994 -1.025486649  0.746669127      low
## 434 -2.809402625  0.499496714 -0.895152022     high
## 35  -1.186967442  1.076441751 -0.982135924 med_high
## 141  0.344005860  1.611376228 -0.927770986 med_high
## 456 -3.349082489  0.766963952 -0.916897998     high
## 215 -0.084824395  2.366165973  0.126908825 med_high
## 96   0.014304949 -0.840640181  0.637939249  med_low
## 464  0.406879058 -0.330912041 -0.253645746     high
## 226  0.277408467 -1.123511291  2.986504601 med_high
## 116 -0.128857540  0.435080520 -0.460232513  med_low
## 137  0.236332772  0.594720652 -0.558089402 med_high
## 202  0.406331382 -0.731412722  0.170400776      low
## 293  0.440615895 -1.113708827  0.583574310      low
## 457 -3.792042783  0.890194931 -1.069119826     high
## 12   0.440615895  0.086392864 -0.394994586  med_low
## 304  0.369746629 -1.091303195  1.148969673  med_low
## 357  0.230636942  0.692745294 -0.514597451     high
## 446 -3.435177145  1.586169891 -1.166976716     high
## 41   0.426595391 -1.494604580  1.344683452      low
## 237  0.348058662 -0.435938444  0.279130654 med_high
## 335  0.358464505 -0.826636661 -0.199280807      low
## 383  0.440615895  1.532956514 -1.221341655     high
## 273  0.419366069 -0.689402161  0.203019739  med_low
## 315  0.427362137 -0.472347596  0.137781813 med_high
## 272  0.440615895 -0.849042293  0.290003641  med_low
## 461 -1.111169093  0.527503755 -0.666819280     high
## 395  0.440615895  0.517701290 -1.069119826     high
## 414 -1.595971828  1.040032598 -0.677692268     high
## 104  0.413998845  0.110198849 -0.351502635  med_low
## 356  0.212125496 -0.991878200 -0.210153795  med_low
## 254  0.440615895 -1.276149663  2.203649484 med_high
## 275  0.440615895 -1.277550015  1.072858759      low
## 102  0.426157250 -0.697804274  0.431352482  med_low
## 484  0.397020891 -0.312707465 -0.079677942 med_high
## 475 -0.044844052  0.768364304 -0.949516961     high
## 133  0.318593697 -0.214682823  0.050797911 med_high
## 350  0.363393588 -0.947066935  0.442225470      low
## 45   0.358354970 -0.434538092 -0.144915868  med_low
## 62   0.234470674  0.250234052 -0.710311231  med_low
## 496  0.401073693  0.692745294  0.061670899  med_low
## 32   0.219683424  0.054184768 -0.873406047 med_high
## 398  0.398992524  1.017626966 -1.525785311     high
## 46   0.440615895 -0.342114857 -0.351502635  med_low
## 372  0.103795196 -0.437338796  2.986504601     high
## 234  0.244000235 -1.218735230  2.801663810 med_high
## 400 -0.202793791  2.424980758 -1.764991042     high
## 80   0.431414939 -0.497553933 -0.242772758  med_low
## 493  0.440615895  0.097595681 -0.264518733  med_low
## 418 -2.511795523  1.958663532 -1.319198544     high
## 98   0.440615895 -1.182326077  1.757856987  med_low
## 337  0.440615895 -0.399529291 -0.329756660      low
## 443  0.427362137  0.551309739 -0.449359525     high
## 245  0.173240505 -0.021434242 -0.536343427  med_low
## 88   0.421447237 -0.589977167 -0.036185991      low
## 302  0.428019349 -0.441539852 -0.057931966      low
## 70   0.440615895 -0.540964846 -0.177534831  med_low
## 429 -2.847301799  1.241683291 -1.253960618     high
## 489  0.420790026  0.757161488 -0.797295133  med_low
## 121  0.355726125  0.240431588 -0.057931966      low
## 14   0.440615895 -0.615183504 -0.231899770 med_high
## 209  0.269960074  0.281041797  0.203019739  med_low
## 161 -0.194469117 -1.001680665  0.485717421 med_high
## 431 -2.992764527  0.698346703 -0.873406047     high
## 9    0.328123258  2.419379350 -0.655946292  med_low
## 473  0.401949974  0.239031236  0.072543887 med_high
## 179  0.378947585 -0.802830676  0.801034065      low
## 4    0.415751408 -1.360170785  1.181588636      low
## 54   0.440615895 -0.591377519  0.094289862      low
## 318  0.440615895  0.460286857 -0.297137697  med_low
## 386  0.440615895  2.542610329 -1.667134152     high
## 16   0.426595391 -0.585776111 -0.286264709 med_high
## 404  0.440615895  0.996621685 -1.547531287     high
## 281  0.335571651 -1.245341918  2.486347165      low
## 79   0.325603949 -0.043839875 -0.144915868      low
## 175  0.400416482 -0.421934923  0.007305960  med_low
## 250  0.406002776 -0.853243349  0.398733519  med_low
## 185  0.375990135  0.185817859  0.420479494  med_low
## 15   0.255720500 -0.335113097 -0.471105500 med_high
## 408 -0.269281649 -0.073247267  0.583574310     high
## 3    0.396035074 -1.207532413  1.322937477      low
## 345  0.342800973 -1.126311996  0.942382906      low
## 93   0.419913745 -0.629187024  0.039924923      low
## 326  0.405345565 -1.060495450  0.224765715  med_low
## 55   0.440615895  0.300646725 -0.394994586      low
## 233  0.320236725 -1.425987330  2.084046619 med_high
## 23   0.440615895  0.849584722 -0.797295133 med_high
## 183  0.410165113 -1.096904603  1.670873085  med_low
## 307  0.440615895 -0.865846518  1.181588636      low
## 336  0.440615895 -0.650192305 -0.155788856      low
##              zn       indus       chas         nox          rm         age
## 1    0.28454827 -1.28663623 -0.2723291 -0.14407485  0.41326292 -0.11989477
## 5   -0.48724019 -1.30558569 -0.2723291 -0.83445805  1.22736204 -0.51067434
## 6   -0.48724019 -1.30558569 -0.2723291 -0.83445805  0.20689164 -0.35080997
## 8    0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.16030692  0.97784057
## 10   0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.39941295  0.61548134
## 19  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.17935407 -1.13592165
## 20  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.79365326  0.03286452
## 21  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.01710355  1.04889141
## 33  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.47626846  0.47693221
## 43  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.16457667 -2.20168412
## 44  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.10480016 -2.20523666
## 48  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.36240845  0.60127117
## 52   0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.24143218 -0.19805068
## 72  -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.46061271 -1.81445709
## 84   0.58468822 -0.91493524 -0.2723291 -1.11061133 -0.16742317 -0.77711496
## 86  -0.48724019 -0.96886832 -0.2723291 -0.91212616  0.49154168 -0.44317605
## 90  -0.48724019 -1.12629463 -0.2723291 -0.56693456  1.13058103 -0.19449814
## 92  -0.48724019 -1.12629463 -0.2723291 -0.56693456  0.17131038  0.18917635
## 99  -0.48724019 -1.20209248 -0.2723291 -0.94664532  2.18520944 -1.12526403
## 107 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.63851899  0.82863383
## 110 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.07918165  0.80376604
## 123 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.46061271  0.86415924
## 142 -0.48724019  1.56744433 -0.2723291  0.59808708 -1.80131441  1.11638970
## 143 -0.48724019  1.23072696  3.6647712  2.72964520 -1.25478633  1.11638970
## 145 -0.48724019  1.23072696 -0.2723291  2.72964520 -1.96641144  1.03823378
## 150 -0.48724019  1.23072696 -0.2723291  2.72964520 -0.97867579  0.93521008
## 158 -0.48724019  1.23072696 -0.2723291  0.43412107  0.93701900  1.02402361
## 166 -0.48724019  1.23072696 -0.2723291  0.43412107 -0.26135768  0.86771179
## 180 -0.48724019 -1.26477147 -0.2723291 -0.57556435  0.98967926 -0.36146759
## 181 -0.48724019 -1.26477147 -0.2723291 -0.57556435  2.10693068  0.52311526
## 187 -0.48724019 -1.26477147 -0.2723291 -0.57556435  2.20086519 -0.53198959
## 192  1.44223095 -1.12192167 -0.2723291 -1.01568364  0.64667596 -1.34196906
## 194  2.08538800 -1.19626187 -0.2723291 -1.32635608  0.73349422 -2.08445025
## 195  2.08538800 -1.19626187 -0.2723291 -1.32635608  0.45453718 -1.76827405
## 200  3.58608778 -1.40907891 -0.2723291 -1.30909650  0.98256301 -1.89261301
## 213 -0.48724019 -0.07970124  3.6647712 -0.56693456 -0.67979324 -0.52488450
## 225 -0.48724019 -0.71961001 -0.2723291 -0.43748771  2.81997903  0.34548818
## 231 -0.48724019 -0.71961001 -0.2723291 -0.43748771 -0.43214771 -0.01687106
## 232 -0.48724019 -0.71961001 -0.2723291 -0.43748771  1.60452335  0.29575260
## 236 -0.48724019 -0.71961001 -0.2723291 -0.41159834 -0.28270643 -0.25133880
## 239  0.79907391 -0.90473168 -0.2723291 -1.09335175  0.27947740 -1.77893168
## 243  0.79907391 -0.90473168 -0.2723291 -1.09335175  0.10441762 -0.55685738
## 246  0.45605682 -0.76917014 -0.2723291 -1.06746238 -0.96728979  0.05773231
## 249  0.45605682 -0.76917014 -0.2723291 -1.06746238  0.21116139 -0.69185396
## 260  0.37030254 -1.04466617 -0.2723291  0.79657225  0.79327073  1.11638970
## 261  0.37030254 -1.04466617 -0.2723291  0.79657225  1.30706406  0.46982713
## 265  0.37030254 -1.04466617 -0.2723291  0.79657225  1.31133381  0.81797620
## 267  0.37030254 -1.04466617 -0.2723291  0.79657225  1.03806976  0.56929830
## 282  0.37030254 -1.13795583 -0.2723291 -0.96476788  0.97260025 -1.11460640
## 284  3.37170210 -1.44697784  3.6647712 -1.32635608  2.33180421 -1.55512156
## 291  2.94293073 -0.90181638 -0.2723291 -1.24005818  0.82031248 -1.44499277
## 297 -0.48724019  0.40569652 -0.2723291 -1.01568364  0.37625841 -0.62435567
## 305  0.92770532 -1.30558569 -0.2723291 -0.71364099  1.35403131 -0.97605728
## 310 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -0.44495696  0.28864751
## 311 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -1.86678392 -1.09329115
## 317 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -0.52750547  0.51956271
## 319 -0.48724019 -0.18027916 -0.2723291 -0.09229612  0.13857563 -0.04884394
## 330 -0.48724019 -1.15107469 -0.2723291 -0.81719847  0.06883637 -1.82511472
## 333  1.01345959 -0.74001712 -0.2723291 -1.00791683 -0.36098520 -1.60840968
## 341 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.45064996 -0.35791505
## 343 -0.48724019 -1.34785757 -0.2723291 -0.31667065  0.36344916 -0.31528455
## 344  1.87100232 -1.07236154 -0.2723291 -0.61008351  0.58547620 -0.43251842
## 347 -0.48724019 -0.98344483 -0.2723291 -0.97253469 -0.55027747 -0.57817263
## 362 -0.48724019  1.01499462 -0.2723291  1.85803641 -0.04787015  0.80021350
## 373 -0.48724019  1.01499462  3.6647712  0.97779784 -0.58301223  0.74692537
## 379 -0.48724019  1.01499462 -0.2723291  1.00368721  0.13572913  0.98139312
## 392 -0.48724019  1.01499462 -0.2723291  1.25395112 -0.33252019  0.49469492
## 394 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.13041866  0.85350162
## 402 -0.48724019  1.01499462 -0.2723291  1.19354259  0.08306887  1.11638970
## 403 -0.48724019  1.01499462 -0.2723291  1.19354259  0.16988713  1.11638970
## 405 -0.48724019  1.01499462 -0.2723291  1.19354259 -1.07261030  0.59771863
## 409 -0.48724019  1.01499462 -0.2723291  0.36508275 -0.95021078  1.04178632
## 411 -0.48724019  1.01499462 -0.2723291  0.36508275 -0.75095575  1.11638970
## 413 -0.48724019  1.01499462 -0.2723291  0.36508275 -2.35780525  1.11638970
## 417 -0.48724019  1.01499462 -0.2723291  1.07272553  0.70787571  0.78955587
## 420 -0.48724019  1.01499462 -0.2723291  1.40928733  0.76765222  0.28154243
## 421 -0.48724019  1.01499462 -0.2723291  1.40928733  0.17984989  1.11638970
## 427 -0.48724019  1.01499462 -0.2723291  0.25289548 -0.63709574 -0.31528455
## 428 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.11760941  0.35969834
## 430 -0.48724019  1.01499462 -0.2723291  1.07272553  0.13572913  0.96007787
## 435 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.10906991  0.93876262
## 436 -0.48724019  1.01499462 -0.2723291  1.59914271  0.49011843  0.92455245
## 448 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.04787015  0.99560328
## 450 -0.48724019  1.01499462 -0.2723291  1.36613839  0.18838939  1.05599649
## 452 -0.48724019  1.01499462 -0.2723291  1.36613839  0.52712294  1.05244395
## 453 -0.48724019  1.01499462 -0.2723291  1.36613839  0.01759936  0.82508129
## 454 -0.48724019  1.01499462 -0.2723291  1.36613839  1.57748160  1.09152190
## 455 -0.48724019  1.01499462 -0.2723291  1.36613839  0.63102020  0.90678974
## 458 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.49619397  0.41653901
## 469 -0.48724019  1.01499462 -0.2723291  0.21837632 -0.51042647  0.08615264
## 477 -0.48724019  1.01499462 -0.2723291  0.51178918  0.28374715  0.88902704
## 478 -0.48724019  1.01499462 -0.2723291  0.51178918 -1.39568810  1.02047107
## 486 -0.48724019  1.01499462 -0.2723291  0.24426569  0.03894811 -0.59238279
## 487 -0.48724019  1.01499462 -0.2723291  0.24426569 -0.24285543  0.39877630
## 490 -0.48724019  2.42017014 -0.2723291  0.46864023 -1.23913058  1.05599649
## 498 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.69829550  0.07194248
## 499 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.37806420 -0.11634223
## 501 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.36667820  0.39522376
## 502 -0.48724019  0.11562398 -0.2723291  0.15796779  0.43888142  0.01865435
## 503 -0.48724019  0.11562398 -0.2723291  0.15796779 -0.23431593  0.28864751
## 504 -0.48724019  0.11562398 -0.2723291  0.15796779  0.98398626  0.79666096
## 506 -0.48724019  0.11562398 -0.2723291  0.15796779 -0.36240845  0.43430172
##               dis        rad         tax     ptratio       black
## 1    0.1400749840 -0.9818712 -0.66594918 -1.45755797  0.44061589
## 5    1.0766711351 -0.7521778 -1.10502160  0.11292035  0.44061589
## 6    1.0766711351 -0.7521778 -1.10502160  0.11292035  0.41016511
## 8    1.0236248974 -0.5224844 -0.57694801 -1.50374851  0.44061589
## 10   1.3283202075 -0.5224844 -0.57694801 -1.50374851  0.32899954
## 19   0.0006920764 -0.6373311 -0.60068166  1.17530274 -0.74137830
## 20   0.0006920764 -0.6373311 -0.60068166  1.17530274  0.37544246
## 21   0.0013569352 -0.6373311 -0.60068166  1.17530274  0.21793086
## 33   0.0925850666 -0.6373311 -0.60068166  1.17530274 -1.35904722
## 43   0.9145880470 -0.7521778 -1.03975408 -0.25660396  0.29241479
## 44   0.9145880470 -0.7521778 -1.03975408 -0.25660396  0.41388931
## 48   0.8996287230 -0.7521778 -1.03975408 -0.25660396  0.39504926
## 52   1.4340327636 -0.6373311 -0.98041997 -0.76469989  0.40852209
## 72   0.7086717651 -0.6373311 -0.61254848  0.34387304  0.22198366
## 84   0.7625253315 -0.6373311 -0.75495035  0.25149196  0.37204687
## 86   0.3051974268 -0.7521778 -0.95668632  0.02053927  0.39022971
## 90  -0.1807194081 -0.8670245 -0.82021787 -0.30279450  0.43141494
## 92  -0.3338269018 -0.8670245 -0.82021787 -0.30279450  0.40392161
## 99  -0.1423950448 -0.8670245 -0.78461740 -0.21041342  0.40370254
## 107 -0.7522605641 -0.5224844 -0.14380900  1.12911220  0.42714307
## 110 -0.5935967501 -0.5224844 -0.14380900  1.12911220  0.37850944
## 123 -0.8111955516 -0.8670245 -1.30675758  0.29768250  0.23458021
## 142 -1.1186927669 -0.6373311  0.17066179  1.26768382  0.44061589
## 143 -1.1746358896 -0.5224844 -0.03107419 -1.73470120  0.44061589
## 145 -1.1630958396 -0.5224844 -0.03107419 -1.73470120  0.44061589
## 150 -1.0777089681 -0.5224844 -0.03107419 -1.73470120 -0.05284012
## 158 -0.9107344185 -0.5224844 -0.03107419 -1.73470120  0.07400163
## 166 -0.7178778639 -0.5224844 -0.03107419 -1.73470120 -1.27623862
## 180 -0.4587728745 -0.7521778 -1.27709053 -0.30279450  0.44061589
## 181 -0.5005640019 -0.7521778 -1.27709053 -0.30279450  0.42593818
## 187 -0.2829652003 -0.7521778 -1.27709053 -0.30279450  0.39384437
## 192  1.2749890302 -0.5224844 -0.06074124 -1.50374851  0.36186010
## 194  1.1514202651 -0.9818712 -0.84988492 -1.31898635  0.40194997
## 195  1.1514202651 -0.9818712 -0.84988492 -1.31898635  0.21935482
## 200  1.8323307009 -0.7521778 -0.03700760 -0.67231881  0.44061589
## 213 -0.0676459148 -0.6373311 -0.77868399  0.06672981  0.37533292
## 225 -0.4277144686 -0.1779443 -0.60068166 -0.48755665  0.31081670
## 231 -0.0586703204 -0.1779443 -0.60068166 -0.48755665  0.23742812
## 232 -0.0586703204 -0.1779443 -0.60068166 -0.48755665  0.21322085
## 236 -0.0679783442 -0.1779443 -0.60068166 -0.48755665  0.21990249
## 239  1.1373157596 -0.4076377 -0.64221553 -0.85708096  0.24903885
## 243  1.5388905012 -0.4076377 -0.64221553 -0.85708096  0.17608842
## 246  1.9755128019 -0.2927910 -0.46421320  0.29768250  0.35550706
## 249  1.9145357480 -0.2927910 -0.46421320  0.29768250  0.19755732
## 260 -0.8473828687 -0.5224844 -0.85581834 -2.51994036  0.38617691
## 261 -0.7992280924 -0.5224844 -0.85581834 -2.51994036  0.39570647
## 265 -0.8856597421 -0.5224844 -0.85581834 -2.51994036  0.34192469
## 267 -0.7893501896 -0.5224844 -0.85581834 -2.51994036  0.30008225
## 282  0.6884410603 -0.5224844 -1.14062207 -1.64232012  0.38946296
## 284  0.9925190015 -0.9818712 -1.24742347 -2.24279713  0.42550004
## 291  0.6276539660 -0.6373311 -0.96855315  0.34387304  0.44061589
## 297  1.0283263992 -0.6373311 -0.70748306 -1.13422420  0.39625414
## 305  0.1077818401 -0.2927910 -1.10502160 -0.02565127  0.40534556
## 310 -0.3288879504 -0.6373311 -0.61848189 -0.02565127  0.43338657
## 311 -0.6058016588 -0.6373311 -0.61848189 -0.02565127 -0.06817505
## 317  0.0966691995 -0.6373311 -0.61848189 -0.02565127  0.37270408
## 319 -0.1246813056 -0.6373311 -0.61848189 -0.02565127  0.42210445
## 330  0.6741465952 -0.6373311  0.12912791 -0.71850935  0.20303408
## 333  1.3514003073 -0.9818712 -0.61848189 -0.71850935  0.06107647
## 341  0.4830471675 -0.5224844 -1.09315478  0.80577843  0.44061589
## 343  1.1738829960 -0.9818712  0.08166062 -1.18041474  0.36459848
## 344  0.9199069177 -0.5224844 -0.22687676 -0.39517558  0.44061589
## 347  2.0033893834 -0.7521778 -0.33367816  0.15911089  0.08692678
## 362 -0.7121315839  1.6596029  1.52941294  0.80577843 -0.06598434
## 373 -1.2658165310  1.6596029  1.52941294  0.80577843 -0.09632559
## 379 -1.1440048928  1.6596029  1.52941294  0.80577843  0.44061589
## 392 -0.7727762084  1.6596029  1.52941294  0.80577843  0.23775673
## 394 -0.9516232374  1.6596029  1.52941294  0.80577843  0.44061589
## 402 -1.0547238481  1.6596029  1.52941294  0.80577843  0.44061589
## 403 -1.0239028917  1.6596029  1.52941294  0.80577843  0.21289224
## 405 -1.0389097056  1.6596029  1.52941294  0.80577843 -0.29808940
## 409 -1.1114268095  1.6596029  1.52941294  0.80577843 -0.46042055
## 411 -1.1312301050  1.6596029  1.52941294  0.80577843 -3.87835651
## 413 -1.0643168114  1.6596029  1.52941294  0.80577843 -3.59148386
## 417 -0.9381835908  1.6596029  1.52941294  0.80577843 -3.67056826
## 420 -0.9502935197  1.6596029  1.52941294  0.80577843 -3.37613768
## 421 -0.9194725633  1.6596029  1.52941294  0.80577843 -0.41540159
## 427 -0.8536040479  1.6596029  1.52941294  0.80577843 -3.63683142
## 428 -0.9175729666  1.6596029  1.52941294  0.80577843 -3.70069044
## 430 -0.8675660836  1.6596029  1.52941294  0.80577843 -3.24173801
## 435 -0.7469416934  1.6596029  1.52941294  0.80577843 -2.80458308
## 436 -0.7932443629  1.6596029  1.52941294  0.80577843 -2.70359163
## 448 -0.7584342534  1.6596029  1.52941294  0.80577843  0.34882541
## 450 -0.7646079427  1.6596029  1.52941294  0.80577843 -0.57466575
## 452 -0.6837801032  1.6596029  1.52941294  0.80577843 -0.01516002
## 453 -0.6776064140  1.6596029  1.52941294  0.80577843  0.31125484
## 454 -0.6374774338  1.6596029  1.52941294  0.80577843  0.21026340
## 455 -0.6168668096  1.6596029  1.52941294  0.80577843 -3.83366615
## 458 -0.4824228534  1.6596029  1.52941294  0.80577843 -3.86849834
## 469 -0.4210658801  1.6596029  1.52941294  0.80577843  0.13216481
## 477 -0.7074775720  1.6596029  1.52941294  0.80577843  0.43305797
## 478 -0.8046419430  1.6596029  1.52941294  0.80577843 -0.07879996
## 486  0.0933923952  1.6596029  1.52941294  0.80577843  0.34992076
## 487 -0.1183176566  1.6596029  1.52941294  0.80577843  0.39439205
## 490 -0.9686246278 -0.6373311  1.79641644  0.75958789 -0.13827757
## 498 -0.4285217972 -0.4076377 -0.10227512  0.34387304  0.44061589
## 499 -0.6581830377 -0.4076377 -0.10227512  0.34387304  0.44061589
## 501 -0.6158695213 -0.4076377 -0.10227512  0.34387304  0.44061589
## 502 -0.6251775451 -0.9818712 -0.80241764  1.17530274  0.38683412
## 503 -0.7159307773 -0.9818712 -0.80241764  1.17530274  0.44061589
## 504 -0.7729186782 -0.9818712 -0.80241764  1.17530274  0.44061589
## 506 -0.6126402069 -0.9818712 -0.80241764  1.17530274  0.44061589
##            lstat        medv    crime
## 1   -1.074498970  0.15952779      low
## 5   -1.025486649  1.48603229      low
## 6   -1.042290874  0.67055821      low
## 8    0.909799859  0.49659041  med_low
## 10   0.622727693 -0.39499459  med_low
## 19  -0.134862757 -0.25364575 med_high
## 20  -0.192277190 -0.47110550 med_high
## 21   1.171665689 -0.97126294 med_high
## 33   2.108501199 -1.01475489 med_high
## 43  -0.958269752  0.30087663  med_low
## 44  -0.730012370  0.23563870  med_low
## 48   0.860787538 -0.64507330  med_low
## 52  -0.451342316 -0.22102678      low
## 72  -0.388326475 -0.09055093  med_low
## 84  -0.720209906  0.03992492      low
## 86  -0.857444405  0.44222547      low
## 90  -0.973673624  0.67055821      low
## 92  -0.623585616 -0.05793197      low
## 99  -1.271948607  2.31237936      low
## 107  0.841182610 -0.32975666  med_low
## 110  0.405673128 -0.34062965 med_high
## 123  0.738956911 -0.22102678  med_low
## 142  3.046737061 -0.88427903 med_high
## 143  1.983869868 -0.99300891 med_high
## 145  2.329756820 -1.16697672 med_high
## 150  1.231880827 -0.77554916 med_high
## 158 -1.129112700  2.04055467 med_high
## 166 -0.398128939  0.26825767 med_high
## 180 -1.066096858  1.59476217      low
## 181 -0.713208146  1.87745985      low
## 187 -1.148717628  2.98650460      low
## 192 -1.115109179  0.86627199      low
## 194 -1.067497210  0.93150992      low
## 195 -1.158520092  0.71405016      low
## 200 -1.133313756  1.34468345      low
## 213  0.472890025 -0.01444002  med_low
## 225 -1.192128541  2.42110924 med_high
## 231 -0.140464165  0.19214675 med_high
## 232 -1.036689465  0.99674784 med_high
## 236 -0.248291271  0.15952779 med_high
## 239 -0.881250390  0.12690883  med_low
## 243 -0.200679302 -0.03618599  med_low
## 246  0.813175569 -0.43848654  med_low
## 249 -0.438739148  0.21389273  med_low
## 260 -0.805631380  0.82278004 med_high
## 261 -0.428936683  1.22508059 med_high
## 265 -0.637589136  1.51865126 med_high
## 267  0.299246373  0.88801797 med_high
## 282 -1.129112700  1.39904839      low
## 284 -1.329363040  2.98650460      low
## 291 -1.305557056  0.64881224      low
## 297 -0.737014131  0.49659041      low
## 305 -0.801430324  1.47515931      low
## 310 -0.375723306 -0.24277276 med_high
## 311 -0.001829314 -0.69943824 med_high
## 317  0.794970993 -0.51459745 med_high
## 319 -0.321109577  0.06167090 med_high
## 330 -0.744015891  0.00730596      low
## 333 -0.675398641 -0.34062965      low
## 341 -0.470947244 -0.41674056      low
## 343 -0.560569774 -0.65594629      low
## 344 -0.766421523  0.14865480      low
## 347  0.002371742 -0.57983538      low
## 362  0.215225251 -0.28626471     high
## 373 -0.528361678  2.98650460     high
## 379  1.545559682 -1.02562788     high
## 392  0.855186130  0.07254389     high
## 394  0.352459751 -0.94951696     high
## 402  1.073641047 -1.66713415     high
## 403  1.072240695 -1.13435775     high
## 405  2.062289582 -1.52578531     high
## 409  1.925055083 -0.57983538     high
## 411 -0.356118378 -0.81904111     high
## 413  3.041135653 -0.50372446     high
## 417  1.839633609 -1.63451519     high
## 420  1.412526239 -1.53665830     high
## 421  0.331454470 -0.63420032     high
## 427  0.425278056 -1.34094452     high
## 428  0.261436868 -1.26483361     high
## 430  1.600173411 -1.41705543     high
## 435  0.352459751 -1.17784970     high
## 436  1.486744897 -0.99300891     high
## 448  0.530304459 -1.07999281     high
## 450  0.932205492 -1.03650086     high
## 452  0.710949871 -0.79729513     high
## 453  0.646533677 -0.69943824     high
## 454  0.572315020 -0.51459745     high
## 455  0.848184370 -0.82991410     high
## 458  0.600322060 -0.98213592     high
## 469  0.766963952 -0.37324861     high
## 477  0.843983314 -0.63420032     high
## 478  1.716402630 -1.14523074     high
## 486 -0.290301832 -0.14491587 med_high
## 487  0.325853062 -0.37324861     high
## 490  1.584769539 -1.68888013  med_low
## 498  0.202622083 -0.46023251 med_high
## 499  0.037380543 -0.14491587  med_low
## 501  0.234830180 -0.62332733  med_low
## 502 -0.417733867 -0.01444002      low
## 503 -0.500354637 -0.21015379      low
## 504 -0.982075736  0.14865480      low
## 506 -0.668396881 -1.15610373      low

Fit the linear discriminant analysis on the train set. Use the categorical crime rate (“crime”) as the target variable and all the other variables in the dataset as predictor variables. Draw the LDA (bi)plot.

## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2376238 0.2722772 0.2500000 0.2400990 
## 
## Group means:
##                   zn      indus        chas        nox         rm
## low       1.14812060 -0.9058553 -0.10828322 -0.9294756  0.3805133
## med_low  -0.09023688 -0.3103016 -0.02178633 -0.5706297 -0.1008021
## med_high -0.39214634  0.2448498  0.27340760  0.3799499  0.1658428
## high     -0.48724019  1.0172187 -0.02879709  1.0571563 -0.4980867
##                 age        dis        rad        tax     ptratio
## low      -0.9599598  0.9984033 -0.6624538 -0.7259014 -0.47889593
## med_low  -0.3476450  0.3256522 -0.5475419 -0.4761340 -0.05630499
## med_high  0.4173480 -0.3518439 -0.4110490 -0.2912043 -0.22916404
## high      0.8188552 -0.8600010  1.6371072  1.5133254  0.77958792
##                black       lstat        medv
## low       0.37865777 -0.74298021  0.49002131
## med_low   0.32561889 -0.18387508  0.02667965
## med_high  0.07620643 -0.04591961  0.19666819
## high     -0.65000902  0.88140303 -0.66087837
## 
## Coefficients of linear discriminants:
##                  LD1         LD2         LD3
## zn       0.091941934  0.82487563 -0.97191792
## indus    0.048657886 -0.17485541 -0.03661729
## chas    -0.085216027 -0.05912733  0.02160071
## nox      0.388873747 -0.74182440 -1.33669281
## rm      -0.084143860 -0.20165378 -0.19913041
## age      0.283695372 -0.31483509 -0.18636568
## dis      0.007025698 -0.32796666 -0.03576485
## rad      3.162267607  1.10414211 -0.19347746
## tax      0.009950956 -0.19391080  0.80153366
## ptratio  0.129069567 -0.08483473 -0.23779991
## black   -0.095773655  0.02211754  0.09997623
## lstat    0.248158882 -0.20383563  0.28350891
## medv     0.228423642 -0.39705787 -0.17994418
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9375 0.0489 0.0137

  • According to the linear description fit you can see how most of the high crime category correlates with the rad (index of accessibility to radial highways).

Save the crime categories from the test set and then remove the categorical crime variable from the test dataset. Then predict the classes with the LDA model on the test data. Cross tabulate the results with the crime categories from the test set.

##   [1] low      low      low      med_low  med_low  med_high med_high
##   [8] med_high med_high med_low  med_low  med_low  low      med_low 
##  [15] low      low      low      low      low      med_low  med_high
##  [22] med_low  med_high med_high med_high med_high med_high med_high
##  [29] low      low      low      low      low      low      low     
##  [36] med_low  med_high med_high med_high med_high med_low  med_low 
##  [43] med_low  med_low  med_high med_high med_high med_high low     
##  [50] low      low      low      low      med_high med_high med_high
##  [57] med_high low      low      low      low      low      low     
##  [64] high     high     high     high     high     high     high    
##  [71] high     high     high     high     high     high     high    
##  [78] high     high     high     high     high     high     high    
##  [85] high     high     high     high     high     high     high    
##  [92] high     med_high high     med_low  med_high med_low  med_low 
##  [99] low      low      low      low     
## Levels: low med_low med_high high
##           predicted
## correct    low med_low med_high high
##   low       12      16        3    0
##   med_low    1      11        4    0
##   med_high   0      10       14    1
##   high       0       0        0   30
  • You can see how the predicted and corrected data tend to agree on the most extreme values (“low” and “high”), in the “med-low” and “med-high” you can see the most discrepance between the correct data and the predicted one.

Reload the Boston dataset and standardize the dataset (we did not do this in the Datacamp exercises, but you should scale the variables to get comparable distances). Calculate the distances between the observations.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1343  3.4625  4.8241  4.9111  6.1863 14.3970
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.2662  8.4832 12.6090 13.5488 17.7568 48.8618
  • You can see how the two methods to calculate the distances between the variables , gives you as result different numbers, but you can also note how the trend between the variations is maintained.

Run k-means algorithm on the dataset. Investigate what is the optimal number of clusters and run the algorithm again. Visualize the clusters (for example with the pairs() or ggpairs() functions, where the clusters are separated with colors):

  • While checking for the optimal number of clusters you should look in the graph when defining the k max = 10 for the k value where the curve varies the most dramatically. That appears to be around k = 2

  • When exploring the pairs using the colors to separate the km clusters, you can see that the variable that better separetes the two clusters (Red Vs Black cluster) would be tax (full-value property-tax rate per $10,000) and rad (index of accessibility to radial highways). An almost complete cluster separation can also be observed when you check the crim (per capita crime rate by town) and black (1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town) variables.

Optional / Bonus sections:

Perform k-means on the original Boston data with some reasonable number of clusters (> 2). Remember to standardize the dataset.

  • For the reasonable amount of clusters (> 2) I select k = 3 as it was the following position in the curve you could observe a dramatically change.

  • Now when you check at the pairs with the clusters divided by colors, since we have more clusters it gets a bit more confusing to read the data. Nonetheless, after careful observation you the variables were you can more easily identify the different clusters (Red, Black and Green) are medv (median value of owner-occupied homes in $1000s) and partialy successful will be age (proportion of owner-occupied units built prior to 1940).

Then perform LDA using the clusters as target classes. Include all the variables in the Boston data in the LDA model. Visualize the results with a biplot (include arrows representing the relationships of the original variables to the LDA solution).

I was trying the following approach but I guess I missed something, further work/learning required in this section ; if the peer review knows how to fix this and explain me what is wrong I highly appreciate it!!

  • k-means clustering km <-kmeans(boston_scaled, centers = 3)

  • linear discriminant analysis lda.fit <- lda(km ~ . , data = boston_scaled$km)

  • print the lda.fit object lda.fit

  • function for the lda biplot arrows lda.arrows <- function(x, myscale = 2, arrow_heads = 0.1, color = “red”, tex = 0.75, choices = c(1,2)){ heads <- coef(x) arrows(x0 = 0, y0 = 0, x1 = myscale * heads[,choices[1]], y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads) text(myscale * heads[,choices], labels = row.names(heads), cex = tex, col=color, pos=3) }

  • target classes as numeric classes <- as.numeric(boston_scaled$km)

  • plot the lda results plot(lda.fit, dimen = 2, col = classes, pch = classes) lda.arrows(lda.fit, myscale = 5)

  • I got stuck in this section as unfortunetely I couldn’t manage to fix the errors in my code

Super-Bonus: Run the provided code for the (scaled) train data that you used to fit the LDA. The code creates a matrix product, which is a projection of the data points.

## [1] 404  13
## [1] 13  3
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:MASS':
## 
##     select
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout
## Warning: 'scatter3d' objects don't have these attributes: 'col'
## Valid attributes include:
## 'type', 'visible', 'showlegend', 'legendgroup', 'opacity', 'name', 'uid', 'ids', 'customdata', 'hoverinfo', 'hoverlabel', 'stream', 'x', 'y', 'z', 'text', 'hovertext', 'mode', 'surfaceaxis', 'surfacecolor', 'projection', 'connectgaps', 'line', 'marker', 'textposition', 'textfont', 'error_x', 'error_y', 'error_z', 'scene', 'xcalendar', 'ycalendar', 'zcalendar', 'idssrc', 'customdatasrc', 'hoverinfosrc', 'xsrc', 'ysrc', 'zsrc', 'textsrc', 'hovertextsrc', 'textpositionsrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule'

Draw another 3D plot where the color is defined by the clusters of the k-means. How do the plots differ?

  • If I performed this correcly it seems that they don’t differ much. Maybe the clusters don’t separate that well the the data in the extreme negative x values (-2 < x < -4 ).